We construct a derived stack \(\mathcal {X}\) of Laurent F-crystals on , where \(\mathcal {O}_K\) is the ring of integers of a finite extension K of \(\mathbb {Q}_p\) . We first show that its underlying classical stack \(^\textrm{cl}\mathcal {X}\) coincides with the Emerton–Gee stack \(\mathcal {X}_\textrm{EG}\) , i.e. the moduli stack of étale \((\varphi ,\Gamma )\) -modules. Then we prove that the derived stack \(\mathcal {X}\) is classical in the sense that when restricted to truncated animated rings, \(\mathcal {X}\) is equivalent to the sheafification of the left Kan extension of \(\mathcal {X}_\textrm{EG}\) along the inclusion from the classical commutative rings to animated rings.
